Eur. Phys. J. B (2018) 91: 179 https://doi.org/10.1140/epjb/e2018-90090-0 THE EUROPEAN PHYSICAL JOURNAL B Regular Article

The family of topological Hall effects for in skyrmion crystals?

B¨orgeG¨obel1, a, Alexander Mook2, J¨urgenHenk2, and Ingrid Mertig1,2 1 Max-Planck-Institut f¨urMikrostrukturphysik, 06120 Halle (Saale), Germany 2 Institut f¨urPhysik, Martin-Luther-Universit¨atHalle-Wittenberg, 06099 Halle (Saale), Germany

Received 22 February 2018 / Received in final form 1 June 2018 Published online 6 August 2018 c The Author(s) 2018. This article is published with open access at Springerlink.com

Abstract. Hall effects of electrons can be produced by an external magnetic field, –orbit coupling or a topologically non-trivial spin texture. The topological Hall effect (THE) – caused by the latter – is commonly observed in crystals. Here, we show analogies of the THE to the conventional Hall effect (HE), the anomalous Hall effect (AHE), and the spin Hall effect (SHE). In the limit of strong coupling between conduction spins and the local magnetic texture the THE can be described by means of a fictitious, “emergent” magnetic field. In this sense the THE can be mapped onto the HE caused by an external magnetic field. Due to complete alignment of electron spin and magnetic texture, the transverse conductivity is linked to a transverse spin conductivity. They are disconnected for weak coupling of electron spin and magnetic texture; the THE is then related to the AHE. The topological equivalent to the SHE can be found in antiferromagnetic skyrmion crystals. We substantiate our claims by calculations of the edge states for a finite sample. These states reveal in which situation the topological analogue to a quantized HE, quantized AHE, and quantized SHE can be found.

1 Introduction in magnetic systems. The HE can appear quantized as quantum Hall effect (QHE) and is due to an external mag- If an is applied to an netic field that couples to the charge of the conduction in the presence of a static magnetic field in perpendicular electrons; the electrons are deflected by a direction, a across the sample is observed. This in a semiclassical picture. The AHE arises due to spin– “Hall effect” of electrons was discovered almost 140 years orbit coupling (SOC) with intrinsic (Berry curvature) ago [1]. In the following and especially in the last 20 years and extrinsic (skew-scattering and side jump) contribu- it has been found that the external magnetic field is not tions [2], all of which deflect electrons with opposite spin mandatory to produce a Hall effect and can be exchanged into opposite directions. In a ferromagnet, the imbalance for example by spin–orbit coupling [2] or a chiral mag- of majority and minority conduction electrons leads to netic texture [3] that produces an effective magnetic field, both spin and charge transport. In non-magnetic materi- that is a “Berry curvature” [4]. The family of Hall effects als the charge accumulation at both sides of the sample is for electrons in was extended by the quantum Hall compensated, leading to a pure spin Hall effect (SHE). effect [5–7], anomalous Hall effect [2], quantum anomalous This paper addresses the THE of electrons in topolog- Hall effect [8,9], spin Hall effect [10,11], quantum spin Hall ically non-trivial magnetic textures s(r) in two dimen- effect [12,13], and topological Hall effect [3,14–24]. sions. The probably most prominent example is the The Hall resistivity skyrmion [25–29]. Its winding is quantified by the topo- logical charge density HE AHE THE ρxy = ρxy + ρxy + ρxy , (1) ∂s(r) ∂s(r) is the off-diagonal element of the resistivity tensor ρ, nSk(r) = s(r) · × , (2) that connects electric field E and J via ∂x ∂y Ohm’s law E = ρJ. It is given by contributions from the ordinary Hall effect (HE), the anomalous Hall effect that integrates to the integer topological charge (or (AHE) and eventually the topological Hall effect (THE) skyrmion number) [30] ? Contribution to the Topical Issue “Special issue in honor of Hardy Gross”, edited by C.A. Ullrich, F.M.S. Nogueira, Z 1 2 A. Rubio, and M.A.L. Marques. NSk = nSk(r) d r. (3) a e-mail: [email protected] 4π xy Page 2 of 7 Eur. Phys. J. B (2018) 91: 179

Fig. 1. Variety of topological Hall effects. (a) In the strong-coupling limit the electron spin aligns with the skyrmion texture and exhibits a topological version of the spin-polarized HE. Only one spin-type of carrier contributes to transport at a certain energy, and spin and charge conductivities are inseparable. This makes the spin-polarized THE a special case of a TAHE. (b) In the weak-coupling limit the local spin is incomplete, leading to an unequal number of spin parallel and antiparallel electrons being deflected into opposite directions. This gives nonzero, uncoupled spin and charge conductivities, i.e., a TAHE. (c) In an antiferromagnetic skyrmion crystal an equal number of spin parallel electrons on the two sublattices are deflected into opposite directions leading to zero charge conductivity. Since the sublattice textures are oriented oppositely this leads to a pure topological spin Hall effect. The large arrows represent the of the conduction electrons as response to an applied electric field in longitudinal direction. Electrons are deflected in transverse direction due to interaction with the texture. They accumulate charge (spheres) and spin represented by small arrows (white: parallel; black: antiparallel and gray levels: partially oriented spins). The colormap represents the z orientation of the magnetic texture. (e) Shows the unit cell of the skyrmion crystal for λ = 10. An arrow represents the magnetic moment si at lattice site i. (d) and (f) are the quantized versions of the topological Hall effects from (a) to (c), respectively. They are observable if the is located in a band gap between two Landau levels. Edge states are represented by the colored lines, where the color represents the spin channel. In the case of spin mixing no quantized version of the TAHE is observed. (g)–(i) of the zero-field band structure (schematic) and position of the Fermi energy for the corresponding scenarios.

The spins of the conduction electrons tend to align the tight-binding Hamiltonian with the magnetic texture, thereby accumulating a Berry [31]. If skyrmions are arranged periodically in a X † X † H = t c cj + m si · (c σci), (4) skyrmion crystal (SkX) this Berry phase can be trans- i i hiji i lated to a Berry curvature in reciprocal space that deflects traversing electrons and leads to nonzero transverse Hall that features nearest-neighbor hopping of conduction elec- coefficients for charge and spin. Skyrmion crystals have † first been observed in non-centrosymmetric B20 bulk- trons (ci creation and ci annihilation operators at site compounds like MnSi [29] and FeGe [32]. They also appear i, amplitude t), and Hund’s coupling (quantified by m), at interfaces, like in Fe/Ir [33]. i.e., the Zeeman interaction of the electrons’ spins with This paper is dedicated to characterize precisely the the fixed magnetic texture {si} (visualized in Fig. 1e). THE and to update the classification of electronic Hall The magnetic texture is thought to be produced by local- effects. The THE is commonly mentioned in line with the ized electrons that are not explicitly featured in the other Hall effects. We think it is more sensible to speak calculation. of topological versions of the conventional, the anomalous To segregate the effect of the magnetic texture, we and the spin Hall effect because the THE manifests itself neglect (i) SOC which would contribute to the AHE differently in dependence on the strength of the Zeeman and (ii) the influence of external magnetic fields that are interaction of electron spin and magnetic texture. We sug- commonly needed to stabilize a SkX phase; they would gest a nomenclature (Fig. 1) for the variety of THEs and contribute to the conventional or quantized Hall effect. Having diagonalized the Hamiltonian, the eigenener- tell which effect can be expected in the different scenarios. gies En(k) and the eigenvectors |un(k)i, both depending In the strong-coupling limit, the Hamiltonians of THE on the wave vector k, allow to calculate the Berry and a spin-polarized QHE can be mapped onto each curvature [4] other (Sect. 4). This locally fully spin-polarized quantized topological Hall effect (QTHE) is the extreme case of a (xy) X hunk|∂kx Hk|umki humk|∂ky Hk|unki quantized topological anomalous Hall effect (QTAHE). Ωn (k) = −2 Im 2 , (Enk − Emk) In the weak-coupling limit spin and charge transport m6=n coefficients are decoupled, what corresponds to the typ- (5) ical anomalous Hall scenario in its topological version (TAHE; Sect. 5). The topological analogue to the spin Hall whose integral over the Brillouin zone (BZ) yields the effect [34–36] (TSHE), i.e., pure spin currents, is found Chern number in antiferromagnetic skyrmion crystals and completes the Z 1 (xy) 2 topological Hall trio (Sect. 6; see Fig. 1). By calculat- Cn = Ωn (k) d k, (6) ing edge states of a finite sample we illustrate when these 2π BZ topological Hall effects show up in their quantized version. of band n and the topological Hall conductivity [2]

e2 1 X Z σ (E ) = − Ω(xy)(k) f(E (k) − E ) d2k. xy F h 2π n n F n BZ 2 Model and methods (7)

In this paper we focus on the topological contribution to Concerning spin transport in a ferromagnet, one veloc- the Hall effect that arises due to the nonzero topological ity vx = ∂kx Hk has to be replaced by the spin current 1 charge density nSk(ri) of the magnetic texture formed by operator Jx = 2 {vx, M}, with M = diag(σz, . . . , σz) localized magnetic moments {si}. Therefore, we consider that accounts for spin-up and spin-down orientation with Eur. Phys. J. B (2018) 91: 179 Page 3 of 7

respect to the global axis M = M ez [37]. case the Zeeman term of the Hamiltonian can be diago- The noncollinear texture of a skyrmion requires to con- nalized by applying a set {Ui} of unitary transformations, sider the local quantization axis (at each site) by means of the operator M = diag(s · σ,..., s · σ). We obtain † 1 n U (si · σ)Ui = σz, (11) the spin Berry curvature i so that the spin at site i points into the z direction S X hunk|Jx|umki humk|∂ky Hk|unki of the local coordinate system. The new electron oper- Ωn,xy(k) = −2 Im 2 , † (Enk − Emk) ators readc ˜ = U c , while the new hopping strength m6=n i i i ˜ † (8) tij = Ui tUj is now a non-diagonal (2 × 2) matrix. The full transformation is discussed in reference [19]. and the spin Hall conductivity for spin polarization In the strong-coupling limit, spin-parallel electrons can- parallel to the local quantization axes not align antiparallel to the magnetic texture and vice versa. For the parallel aligned electrons, only the (1, 1) Z element S e 1 X S 2 σxy(EF) = Ωn,xy(k) f(En(k) − EF) d k. 4π 2π BZ  θ θ θ θ  n t˜(1,1) = t cos i cos j + sin i sin j exp[−i(φ − φ )] , (9) ij 2 2 2 2 i j (12) 3 Influence of the Hund’s coupling on the of the hopping is relevant, which depends on the azimuthal band structure θi and the polar angle φi of the local magnetic moment at site i. It can be transformed into an effective hopping Before presenting results for the transverse charge and strength spin conductivities, the effect of the Hund’s coupling on the electron spin and on the band structure is prerequisite. (eff) θij t = t eiaij cos , (13) The strength of the interaction is quantified by m/t. ij 2 Aiming at the simplest, yet most distinct manifestation of topological properties of the SkX in band structure with cos θij = si · sj the angle between the moments at and transport coefficients, we consider only s-electrons sites i and j; the phase is given by on a square structural lattice (lattice constant a). The respective zero-field band structure (band structure of the − sin(φi − φj) aij = arctan . (14) structural lattice for m = 0) has one band, θi θj cos(φi − φj) + cot 2 cot 2 E (k) = 2t [cos(k a) + cos(k a)] , (10) 0 x y With these substitutions the Hamiltonian becomes with a maximum at 4t, a minimum at −4t, and two saddle X θij H = t cos exp (ia ) d†d , (15) points at EVHS = 0 (van Hove singularity, VHS). On top 2 ij i j of this, it is spin-degenerate. hiji The square magnetic unit cell of the SkX, with a period of λ · a, leads to back-folding of the bands into which describes fully spin-aligned electrons (equivalent to 2 † the magnetic Brillouin zone and thus to n = λ bands. spinless electrons; creation di and annihilation di). The A nonzero coupling m breaks time reversal symmetry, constant Zeeman term has been dropped. The cosine in and the spin degeneracy is lifted. This is accompanied (13) merely scales the hopping strength and converges to by a shift of the bands to lower (spin antiparallel to the 1 for large skyrmions (λ → ∞); the THE has then been magnetic texture) and higher (spin parallel to the mag- mapped onto the QHE described by the Hamiltonian netic texture) energies. For m > 4t (half the band width X of E0), the band structure is separated into two blocks, H = t exp(ib ) d†d . (16) the electron spin is almost completely aligned with the ij i j magnetic texture, and small band gaps show up. These hiji findings are best understood in the strong-coupling limit In the quantum Hall effect spinless electrons interact with m/t → ∞. a uniform magnetic field B, that is oriented out of plane (spin-polarized electrons can be described likewise). This 4 Quantized topological Hall effect in the field couples to the charge, rather than to the spin of the conduction electrons and affects their kinetic energy. strong-coupling limit In the tight-binding description this leads to complex hopping amplitudes with phase factors 4.1 Transformation to the emergent field Z In the limit m/t → ∞ the electron spin completely aligns bij = e/~ A(r) · dl, (17) with the texture to minimize the system’s energy. In this ri→rj Page 4 of 7 Eur. Phys. J. B (2018) 91: 179

Fig. 2. Charge and spin Hall conductivity for different m/t exact Hamiltonian (4); the numerical results are only regimes. (a) m/t = 5, (b) m/t = 3 and (c) m/t = 1. Red rep- interpreted by means of the emergent field. 2 resents the charge conductivity in units of σ0 = e /h, black The transverse conductivity (red in Fig. 2a) depends S is the spin Hall conductivity in units of σ0 = e/(4π). The strongly on the position of the Fermi energy. The two upper panel shows the density of states (DOS) of the zero-field blocks in the band structure, easily recognized in the band shifted by ±m. Additive and subtractive superposition spectrum, show opposite sign, which is explained by the of the integrated DOS ( character is respected by sign) opposite spin alignment with respect to the magnetic tex- yields the transparent red and black curves, respectively. They ture, giving emergent fields of opposite sign. This leads to approximate the corresponding conductivities well, if the spin deflection of electrons into opposite transverse directions is aligned with the texture. For small m it still captures in a semi-classical picture. It is sufficient to discuss only qualitative features. The skyrmion size is λ = 8. the upper block. Interpreting the bands as dispersive Landau levels with a Chern number of +1 tells that the conductivity decreases whose phase is determined by the vector potential A, with stepwise upon increasing EF. Due to the inhomogeneity of B = ∇ × A. The above integral is along the hopping path. the emergent field, the bands are “deformed” and have a The phase aij for the topological Hall effect can be iden- nonzero band width. The spectrum consists therefore of tified with the phase bij from the quantum Hall effect. energy regions in which σxy is quantized (insulating sys- The local skyrmion density nSk(r) can therefore be iden- tem; QTHE; Fig. 1d) or not quantized (metallic system; tified with a fictitious magnetic field that is collinear but THE; Fig. 1a). inhomogeneous. The energies of the Landau levels are determined by the Application of the unitary transformations {Ui} is constraint that, like for the QHE, for each Landau level exact; however, taking only a diagonal element is strictly the number of states is identical (Onsager’s quantization m valid only in the strong-coupling limit /t → ∞. In this scheme). At these energies, the corresponding Fermi line limit the Hund’s coupling can be expressed as an interac- of the zero-field band structure encloses an area of tion of the skyrmion’s emergent field with the charge of the conduction electrons. Considering the (2, 2) element   instead of the (1, 1) element, one would have described the 1 ζi = ζ0 i − , i = 1, 2, . . . , n, (18) lower block of bands, in which the electron spin is aligned 2 antiparallel to the magnetic texture. In that case, the emergent field as well as the topological Hall conductivity in the Brillouin zone; ζ is the area of the (smaller) change sign. 0 magnetic Brillouin zone. The curvature of these Fermi Describing the QHE on a lattice restricts the magnetic lines dictates both the fermion character and the sign of field to discrete values, so that the magnetic flux per mag- the transverse conductivity: electron-like (hole-like) orbits netic unit cell is an integer multiple of the flux quantum give a negative (positive) sign for energies below (above) Φ = h/e. Therefore, a large magnetic unit cell has to 0 E . The sign of the transverse conductivity changes at be considered to account for the appropriate phase b . VHS ij the VHS [22,23,38–40], because it separates electron- from The magnetic flux per plaquette then reads Φ = p/q Φ ; p 0 hole-like Fermi lines. The particular Landau level closest and q are coprime integers [6]. The emergent field of the to the VHS thus carries a large Chern number of 1 − n skyrmion induces an integer number of flux quanta per which compensates the Chern number of all other bands unit cell as well. Both systems differ by the fact that the (for details see Ref. [23]). The zero-field band structure emergent field is not homogeneous; still one can find a dictates therefore the shape of σ (E ): different lattices quantum Hall system with a magnetic field equal to the xy F lead to different spectra (cf. a triangular lattice in Ref. [22] average emergent field of the SkX. The average flux per and a honeycomb lattice in Ref. [23]). plaquette then reads Φ = NSk/n Φ = p/q Φ . 0 0 Based on the above observations we developed an easy This equivalence of THE and QHE in the limit m/t → ∞ way to predict σ (E ) (see Ref. [23]): the DOS is inte- allows to carry over all the known results from one effect to xy F grated up to the Fermi energy and shifted at VHSs, the the other. Bands in the THE can, for example, be inter- result is the transparent red curve in the background of preted as dispersive Landau levels which carry a Chern Figure2a. number of +sign(NSk)[−sign(NSk)] for the upper [lower] block. One result is a quantized transverse conductivity, provided the lies within a band gap. Fur- 4.3 Edge states and higher order skyrmions thermore, σxy behaves peculiarly for Fermi energies near VHSs, as is discussed in the next section. The bulk-boundary correspondence connects the sum of Chern numbers of the occupied bands with the number 4.2 Conductivity of topologically protected edge states [41,42], the latter being computed by Green function renormalization for the The analogy of THE and QHE holds strictly speaking semi-infinite system [43,44]. in the limit m/t → ∞. Nevertheless, the emergent-field Figure3a illustrates the edge states of a skyrmion crys- description is reasonable as long as block separation tal in the strong-coupling limit m/t = 900. Due to the large in the band structure is given (m ≥ 4t). We empha- degree of spin polarization we show only the lower block. size that the computations were performed with the The number of edge states changes by 1 for high and low Eur. Phys. J. B (2018) 91: 179 Page 5 of 7

Fig. 3. Edge states of skyrmion crystals. (a) Skyrmion crystal and (b) manipulated SkX in the strong-coupling limit m/t = 900 and for NSk/n = 1/36. (c) Quantum Hall system for p/q = 1/36. (d) SkX with NSk/n = 3/64 and (e) quantum Hall system with p/q = 3/64. Gray: bulk states; red: edge states. energies. The edge states change from left to right prop- In the strong-coupling limit, the THE becomes a spin- agation at EVHS, which complies with the sign change of polarized quantized topological Hall effects, which is the conductivity. related to a spin-polarized QHE brought about by a mag- In Figure 3b the system has been manipulated by netic field (Fig. 1a). The spin-polarized QHE is also the renormalizing the hopping (t → t/ cos(θij/2) according to extreme case of a QAHE. Therefore the spin-polarized Eq. (13)). Additionally, the skyrmion texture has been THE can also be labeled TAHE (Fig. 1a) and the quan- tilted so that the emergent field becomes almost homo- tized version, for Fermi energies located in a band gap, can geneous. This way, the topological charge of a skyrmion also be labeled QTAHE (Fig. 1d). The Hall conductivity remains invariant but the SkX mimics a quantum Hall sys- and the transverse spin conductivity of both compared tem with dispersionless Landau levels (cf. Figs. 3b and 3c) systems agree well. even better. The dispersion relations of edge states in the Note, that we calculate a spin conductivity in a local quantum Hall system and the SkX are very similar, espe- coordinate system. The measured spin accumulation (in cially their slope. Keep in mind that the SkX magnetic a global system) is determined by this quantity but the unit cell comprises λ × λ sites, while for the QH system it texture at the sample’s edge has to be taken into account comprises n × 1 sites (with n = λ2 equal to the number of (cf. accumulated spins in Fig. 1a). in the magnetic unit cell). The ΓX direction of the Commonly, in an AHE scenario charge and spin trans- QH system is λ = 6 times as long as that of the SkX. port are (strictly) uncoupled. This happens in the weak- The equivalence of the spin polarized QTHE and the coupling limit, in which complete spin-polarization is not QHE holds also for larger skyrmion numbers. The ratio given for all Fermi energies. NSk/n dictates the ratio p/q for the related quantum Hall system. Figures 3d and 3e show the results for the ratio 3 5 Topological anomalous Hall effect in the /64, i.e., for a skyrmion with NSk = 3. The expansion weak-coupling limit 3 1 = , (19) 1 In materials with weaker Hund’s coupling m, the sepa- 64 21 + 3 ration into non-overlapping blocks of bands is not given, tells that the band structure shows 21 bundles that are hence the emergent-field picture becomes invalid. connected by surface states like in the case p/q = 1/21 In collinear ferromagnets the Hall effect can be treated (Refs. [45,46]). Each of these bundles consists of three well in a two-channel model, where each channel deals subbands with a total Chern number of −1, except for with one kind of either spin parallel or spin antiparallel the bundle at the VHS, which consists of four bands with electrons. This picture breaks down if the two spin species a total Chern number of +20, compensating the other hybridize, for example, due to spin–orbit coupling or, as Chern numbers. This behavior is well understood for the in our case, due to a noncollinear magnetic texture. When QHE and can be carried over to the topological version. the electron spin does not follow the texture adiabatically, The Diophantine equation [46–48] tells the Chern num- complete local spin polarization, i.e., parallel or antiparal- bers of the subbands. Furthermore, the essential physics lel alignment, is not given. Still, we utilize the two channel are represented by Hofstadter butterflies [6]; such figures model and compare its prediction to the actual results. depict the band energies versus magnetic flux, which is Figures 2b and 2c show charge and spin conductivities characterized by the ratio p/q for the QH case and can for m/t = 3 and m/t = 1, respectively. Both conductivities now be understood as the magnetic flux of the emergent change sign at the energy of the zero-field VHS EVHS = N field given by Φ = Sk/n Φ0. ±m. For energies, at which the two zero-field bands do not overlap (|E| > 4t − m) both curves are well explained by 4.4 Spin conductivity the two-channel model, because the electron spin is almost completely aligned with the magnetic texture. The transverse spin conductivity of the SkX in the strong- For energies at which the blocks overlap, the two- coupling limit is shown in Figure2 (panel a; black curve). channel picture describes qualitatively the spectrum but Charge and spin transport appear inseparably linked, fails to predict the amplitude, especially for small m/t. which is attributed to the high degree of spin polariza- This is due to the fact that the spin is locally not aligned tion. In the case of non-overlapping blocks (m ≥ 4t) the parallel or antiparallel to the texture. The bands become results are well resembled by the emergent field picture. dispersive, and the eigenstates of the Hamiltonian show S The spin conductivity is given by σxy = −~/2e σxy for all contributions from both spin species. Fermi energies in the upper block. For the lower block the For m < 4t the charge conductivity can exceed the spin charge conductivity is sign-reversed but the spin conduc- Hall conductivity in magnitude (in units of their respec- tivity is not. Due to the opposite sign of the emergent field, tive quanta) if spin-antiparallel hole-like states hybridize electrons with opposite spins are deflected into opposite with spin-parallel electron-like states for energies |EF| < m m directions. 2t − |m − 2t| (cf. |EF| < t for /t = 1 and /t = 3 in Page 6 of 7 Eur. Phys. J. B (2018) 91: 179

Figs. 2b and 2c). In the two-channel approximation, then, (arbitrarily labeled as “up”) in one sublattice are inter- electrons with opposite spins are deflected into the same preted as antiparallel spins in the other sublattice (also direction and the spin conductivity can drop to zero. “up”). For the calculation of the spin Hall conductivity A B For m < 2t the spin conductivity can exceed the this is respected by a modified M = diag(s1 · σ, −s1 · A B charge conductivity. This can happen for −4t + m < σ,..., sn · σ, −sn · σ) that enters equation (8) via Jx. EF < −m, where spin-parallel electron-like states mix The results can be understood as two copies of with spin-antiparallel electron-like states, and for m < Figure 1a. Locally spin-parallel electrons from sublattice EF < 4t − m, where spin-parallel hole-like states mix A (spin up) are deflected to the left, and parallel electrons with spin-antiparallel hole-like states. In the two-channel from sublattice B (spin down because of the reversed mag- approximation, electrons with opposite spins are deflected netization) are deflected to the right. This leads to a zero into opposite directions (this corresponds to the scenario charge Hall conductivity and a nonzero spin Hall conduc- S depicted in Fig. 1b). This increases the spin conductiv- tivity, i.e., a TSHE. σxy(EF) looks just like for the SkX, ity and decreases the charge conductivity compared to but with twice the magnitude because of the two sub- the case of non-overlapping zero-field bands. Due to the lattices. Within the band gaps, the TSHE is quantized missing alignment the actual transverse conductivities are (QTSHE, Fig. 1f). always reduced in magnitude compared to the results In real materials, nearest-neighbor hopping, i.e., inter- expected for the two-channel model. sublattice hopping, is allowed. Nevertheless, the THE in The scenario of mixed zero-field blocks corresponds to an AFM-SkX vanishes, in contrast to a topological spin S the AHE in the stricter sense (cf. Fig. 1b), i.e., where Hall effect, as shown in reference [34]. σxy(EF) appears both types of spin carriers dominate the transport effects. modified by the altered zero-field band structure. In this case the emergent field interpretation breaks down and bands do not behave like Landau levels. They become dispersive and band gaps disappear. Therefore, edge states are visible but are always superposed by bulk bands and 7 Conclusion the quantized version of the TAHE for mixed carriers cannot be observed. We have analyzed the electron transport in different coupling regimes in conventional and antiferromagnetic For Fermi energies |EF| > 4t − m, where only one spin species is present, the local spin alignment is given and skyrmion crystals. bands become merely flat. This QTAHE is again equiv- In the strong-coupling limit, the topological equivalent alent to a spin-polarized version of the QHE as in the to a fully spin-polarized QHE is found and explained in the strong-coupling limit (cf. Sect. 4). emergent-field picture. Because the electron spin follows the texture adiabatically, charge and spin transport are twined and it can therefore also be seen as a special case of a TAHE or a QTAHE. 6 Topological spin Hall effect in In the weak-coupling limit, spin-mixing terms in the antiferromagnetic skyrmion crystals Hamiltonian prevent the interpretation by means of the emergent field picture. The charge conductivity is no The nontopological AHE in a ferromagnet arises due to longer quantized and the spin conductivity is not propor- SOC and a net magnetization. In a SkX, the effect of the tional to the charge conductivity. Here, one is confronted SOC is taken over by the magnetic texture, which also with a topological analogue to the typical anomalous Hall induces the block separation. Therefore, a pure topologi- effect: an unequal number of electrons with their spin cal spin Hall effect – the number of spin-parallel electrons aligned parallel and antiparallel to the magnetic texture is identical to the number of spin-antiparallel electrons at propagate into opposite transverse directions, resulting in a decoupling of charge and spin conductivities. EF – could only be achieved when the texture is removed. Consequently, all transverse charge and spin transport An extreme version of the anomalous Hall effect is phenomena would vanish as well. the spin Hall effect, with identical numbers of electrons The topological analogue of the SHE [34–36] has been with spin parallel and antiparallel to the magnetic tex- established in antiferromagnetic (AFM) skyrmions [49–51] ture. This effect is not possible in a skyrmion crystal, and antiferromagnetic SkXs [34]. These textures consist of but in an antiferromagnetic skyrmion crystal. We find a two sublattices or layers with conventional skyrmions, but TSHE or QTSHE, depending on the position of the Fermi with opposite magnetic moments in the two sublattices. energy. Hence, the topological charge and the emergent field of We have shown that topologically nontrivial magnetic each sublattice have opposite signs, leading to a zero total textures lead to a topological Hall effect, which is viewed topological charge and a vanishing THE. not as a separate Hall effect but as the topological ver- In this section we briefly review the results of sion of well-known Hall effects previously discussed in reference [34] to complete the family of topological Hall ferromagnets or conventional metals. effects. First, we consider only hopping within a sublat- tice. The AFM-SkX then consists of two non-interacting SkXs on the two sublattices. The conduction electrons This work is supported by SPP 1666 and SFB 762 of Deutsche are deflected into opposite directions depending on the Forschungsgemeinschaft (DFG). Open access funding provided sublattice on which they “live” (Fig. 1c). Parallel spins by Max Planck Society. Eur. Phys. J. B (2018) 91: 179 Page 7 of 7

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